Particle pressure in sheared Brownian suspensions

Abstract

The isotropic contribution of the particle phase to the bulk stress, or the particle pressure, is studied for Brownian hard sphere suspensions in computationally simulated shear flow. The particle pressure is mechanically defined as the negative mean normal stress exerted by the particles, i.e., Π = − ( 1 ∕ 3 ) [ Σ 11 + Σ 22 + Σ 33 ] for a viscometric flow where 1, 2, and 3 refer to the flow, velocity gradient, and vorticity directions, respectively. Analysis is provided to relate the particle pressure to the equilibrium osmotic pressure and to show the relation of Π to particle migration phenomena. Utilizing existing hydrodynamic functions and simulating the flow by the Stokesian Dynamics technique, the particle pressure is evaluated for particle volume fractions in the range 0.1 ⩽ ϕ ⩽ 0.52 for monodisperse spherical particles. The relative strength of Brownian to shearing motion is given by the Péclet number Pe = γ ̇ a 2 ∕ D 0 , where γ ̇ is the shear rate of a simple shear flow, a is the spherical particle radius, and D 0 = k T ∕ 6 π η a with k T the thermal energy and η the suspending fluid viscosity. For each ϕ , the range 0.1 ⩽ Pe ⩽ 1000 has been studied. The particle pressure at Pe = 0.1 , where it is given predominantly by a Brownian contribution, is found to approach the exact results for the osmotic pressure of an equilibrium hard-sphere dispersion, Π = n k T [ 1 + 4 ϕ g ( 2 a ) ] , where n is the particle number density and g ( 2 a ) is the pair distribution function evaluated at contact. The hydrodynamic contribution to Π grows with Pe and dominates the Brownian contribution at Pe > 10 . The particle pressure scales as η γ ̇ at elevated Pe . The relative contributions to Π of Brownian and hydrodynamic stress are similar as a function of Pe to the normal stress differences of the suspension, N 1 = Σ 11 − Σ 22 and N 2 = Σ 22 − Σ 33 , but, for ϕ ⩾ 0.2 , Π exceeds ∣ N 1 ∣ and ∣ N 2 ∣ for all Pe .